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In probability theory and statistics, the marginal distribution of a subset of a collection of random variables is the probability distribution of the variables contained in the subset. It gives the probabilities of various values of the variables in the subset without reference to the values of the other variables. This contrasts with a conditional distribution, which gives the probabilities contingent upon the values of the other variables. The term marginal variable is used to refer to those variables in the subset of variables being retained. These terms are dubbed "marginal" because they used to be found by summing values in a table along rows or columns, and writing the sum in the margins of the table.〔Trumpler and Weaver (1962), pp. 32–33.〕 The distribution of the marginal variables (the marginal distribution) is obtained by marginalizing over the distribution of the variables being discarded, and the discarded variables are said to have been marginalized out. The context here is that the theoretical studies being undertaken, or the data analysis being done, involves a wider set of random variables but that attention is being limited to a reduced number of those variables. In many applications an analysis may start with a given collection of random variables, then first extend the set by defining new ones (such as the sum of the original random variables) and finally reduce the number by placing interest in the marginal distribution of a subset (such as the sum). Several different analyses may be done, each treating a different subset of variables as the marginal variables. ==Two-variable case== Given two random variables ''X'' and ''Y'' whose joint distribution is known, the marginal distribution of ''X'' is simply the probability distribution of ''X'' averaging over information about ''Y''. It is the probability distribution of ''X'' when the value of ''Y'' is not known. This is typically calculated by summing or integrating the joint probability distribution over ''Y''. For discrete random variables, the marginal probability mass function can be written as Pr(''X'' = ''x''). This is : where Pr(''X'' = ''x'',''Y'' = ''y'') is the joint distribution of ''X'' and ''Y'', while Pr(''X'' = ''x''|''Y'' = ''y'') is the conditional distribution of ''X'' given ''Y''. In this case, the variable ''Y'' has been marginalized out. Bivariate marginal and joint probabilities for discrete random variables are often displayed as two-way tables. Similarly for continuous random variables, the marginal probability density function can be written as ''p''''X''(''x''). This is : where ''p''''X'',''Y''(''x'',''y'') gives the joint distribution of ''X'' and ''Y'', while ''p''''X''|''Y''(''x''|''y'') gives the conditional distribution for ''X'' given ''Y''. Again, the variable ''Y'' has been marginalized out. Note that a marginal probability can always be written as an expected value: : Intuitively, the marginal probability of ''X'' is computed by examining the conditional probability of ''X'' given a particular value of ''Y'', and then averaging this conditional probability over the distribution of all values of ''Y''. This follows from the definition of expected value, i.e. in general : 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Marginal distribution」の詳細全文を読む スポンサード リンク
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